Tarski's Theorem on Intuitionistic logic, for polyhedra

0.27 MB·

by  Nick Bezhanishvili

TARSKI’S THEOREM ON INTUITIONISTIC LOGIC, FOR
POLYHEDRA
NICKBEZHANISHVILI,VINCENZOMARRA,DANIELMCNEILL,
ANDANDREAPEDRINI
7
1
0
2 Abstract. In1938,Tarskiprovedthataformulaisnotintuitionisticallyvalid
if, and only if, it has a counter-model in the Heyting algebra of open sets of
n some topological space. In fact, Tarski showed that any Euclidean space Rn
a withn>1suffices,asdoes e.g.theCantor space. Inparticular,intuitionistic
J
logic cannot detect topological dimension in the frame of all open sets of a

8 Euclidean space. By contrast, we consider the lattice of open subpolyhedra
1 ofagivencompact polyhedron P ⊆Rn,provethat itisalocallyfiniteHeyt-
ing subalgebra of the (non-locally-finite) algebra of all open sets of Rn, and

] show that intuitionistic logic is able to capture the topological dimension of
O
P through the bounded-depth axiom schemata. Further, we show that intu-
L itionistic logic is precisely the logic of formulæ valid in all Heyting algebras
. arising from polyhedra in this manner. Thus, our main theorem reconciles
h through polyhedral geometry two classical results: topological completeness
t inthestyleofTarski,andJa´skowski’stheoremthatintuitionisticlogicenjoys
a
m the finite model property. Several questions of interest remain open. E.g.,
whatistheintermediatelogicofallclosedtriangulablemanifolds?
[

1 1. Introduction
v
4 If X is any topologicalspace,the collectionO(X) of its open subsets is a (com-
9 plete) Heyting algebra whose underlying order is given by set-theoretic inclusion.
0 Onecantheninterpretformulæofintuitionistic logicinto O(X)byassigningopen
5
sets to propositional atoms, and then extending the assignment to formulæ using
0
the operations of the Heyting algebra O(X). A formula is true under such an in-
.
1
terpretation just when it evaluates to X. In 1938,Tarski([33], English translation
0
in [34]) proved that intuitionistic logic is complete with respect to this semantics.
7
1 Moreover, Tarski showed that one can considerably restrict the class C of spaces
: under consideration without impairing completeness. In particular, one can take
v
C:={X | X is metrisable},andevenC:={R}orC:={2N},where2N denotesthe
i
X Cantor space. Tarski’s result opened up a research area that continues to prosper
r to this day. Immediate descendantsof[33]arethe threeseminalpapers[22,23,24]
a
by McKinsey and Tarski; [23, §3] offers a different proof of the main result of [33]
in the dual language of closed sets and co-Heyting algebras. For an exposition of
the different themes in spatial logic we refer to [2].
Intuitionistic logic has the finite model property. In 1936 Ja´skowski sketched a
proof of this fact [17]; the first detailed exposition of the result1 seems to be [29,
Theorem5.4](seealso[10,Theorem2.57]). Algebraically,thefinitemodelproperty
Keywordsandphrases. Intuitionisticlogic;topologicalsemantics;completenesstheorem;finite
modelproperty;Heytingalgebra; locallyfinitealgebra;polyhedron; simplicialcomple; triangula-
tion;PLtopology.
2010MathematicsSubjectClassification. Primary: 03B20. Secondary: 06D20;06D22;55U10;
52B70;57Q99.
1Though not exactly of the proof sketched by Ja´skowski: cf. [29, Lemma 5.3 and footnote
(16)].
1
2 N.BEZHANISHVILI,V.MARRA,D.MCNEILL,ANDA.PEDRINI
mayberephrasedintothestatementthatthereexistsasetoffiniteHeytingalgebras
thatgeneratestheequationalclass(orvariety)ofallHeytingalgebras. Analgebraic
proof of this result was first obtained by McKinsey and Tarski [21, 22] (see [5] for
a discussion of this proof and a comparison with the model-theoretic method of
filtration). Ja´skowski’s proof shows that, in fact, there is a countable, recursively
enumerable2 such set.
Recall that an algebraic structure is locally finite if its finitely generated sub-
structures are finite. The Heyting algebras O(R) and O(2N) are very far from
being locally finite. For example, [23, Theorem 3.33] shows that any Heyting alge-
brafreelygeneratedbyafinitesetembedsintobothO(R)andO(2N),andalready
the Heyting algebra freely generated by one element (the Rieger-Nishimura lattice
[28, 26]) is of course infinite. Thus, while counter-models to formulæ that are not
intuitionistically provable always exist in O(R), they are not automatically finite:
one has to pick the open sets to be assigned to atomic formulæ with extra care in
order to exhibit a finite counter-model such as the ones guaranteed by the finite
model property, see e.g., [6].
OurmainresultprovidesatheoreminthestyleofTarskithathastheadvantage
of using locally finite Heyting algebras of opens sets only, and hence affords at the
sametime theadvantagesofJa´skowski’stheorem. Ourresultexposesandexploits,
we believe for the first time, the connection between intuitionistic logic and the
classicalPL(=piecewiselinear)categoryofcompactpolyhedrainEuclideanspaces
[30, 20]. The needed background is recalled in Section 2, to which the reader is
referred for all unexplained notions in the rest of this Introduction. To state our
results we prepare some notation.
For each n ∈N := {0,1,2,...} and each (always compact) polyhedron P ⊆Rn,
we write Sub P for the collection of subpolyhedra of P — i.e., polyhedra in Rn
c
contained in P. We set
Sub P :={O ⊆P |P \O∈Sub P},
o c
where\isset-theoreticdifference. MembersofSub P arecalledopen (sub)polyhedra
o
(ofP). ItisastandardfactthatSub P isadistributivelatticeunder set-theoretic
o
intersections and unions, and hence a sublattice of3 O(P). In Section 3 we prove
thatSub P is,infact,aHeytingsubalgebraofO(P). Inthesamesectionweprove
o
that, unlike O(P), Sub P is always locally finite. The proof provides one of the
o
key insights of the presentpaper: local finiteness essentially amounts to the Trian-
gulation Lemma of PL topology, and thus reflects algebraically a crucial tameness
property of polyhedra as opposed to general compact subsets of Rn.
Further tameness properties of polyhedra emerge from their dimension theory,
which is far simpler than the dimension theory of general metric spaces. All stan-
dard topological dimension theories agree on polyhedra [16, 27]. In fact, an ele-
mentary notion of dimension is available for every nonempty polyhedron ∅ =6 P,
in that dimP 6 d holds if, and only if, any d+2 distinct points of P are affinely
dependent. In Section 4, we establish a fundamental connection between the topo-
logicaldimension of P and the structure of Sub P: the latter lies in the variety of
o
Heyting algebras of bounded depth d if, and only if, dimP 6 d. Recall that the
bounded-depth axiom schemata (see, e.g., [10, Section 2.5]) are inductively defined
2Each finite Heyting algebra being presented, e.g., by the finite multiplication tables for its
operations. Ja´skowski’s theorem yields at once the decidability of intuitionistic logic. More is
known: theproblemofdecidingwhetheraformulaisintuitionisticallyprovableispspace-complete
[32].
3Here and throughout, P is always equipped with the subspace topology inherited from the
EuclideantopologyofRn.
TARSKI’S THEOREM AND POLYHEDRA 3
as follows, over the countably infinite set {α ,...,α ,...} of propositions:
0 n
(α ∨¬α ) if d=0, and
bd := 0 0
d
((αd∨(αd →bdd−1)) if d>1.
IfnowP isanyfamilyofpolyhedra,wewriteLogP fortheextensionofintuition-
istic logic determined by P, namely, the unique intermediate logic corresponding
to the variety of Heyting algebras generated by the collection of Heyting algebras
{Sub P |P ∈P}.
o
Given d ∈ N, let us denote by P the set of all polyhedra of dimension less
d
than or equal to d. Consider any finite poset A of depth d ∈ N. In Section 5,
using Alexandrov’s notion of nerve [3], we construct a polyhedron P of dimension
dsuchthattheHeytingalgebraofuppersetsofAembedsintotheHeytingalgebra
Sub P. This leads to our main result:
o
Theorem. For each d ∈ N, LogP is intuitionistic logic extended by the axiom
d
schema bd . Hence, the logic Log P of all polyhedra is intuitionistic logic.
d d∈N d
We prove the theorem in SectioSn 6. Our proof is self-contained to within the
standard facts from PL topology and Heyting algebras recalled in Section 2.
ReturningtoTarski’stheorem,letusconsiderEuclideanspacesRN andRn with
N > n ∈ N. In line with the compact setting of the present paper, let us in fact
confineattentionto theirunitcubes [0,1]N and[0,1]n. ThenTarski’sresultsshow,
inter alia, that the Heyting algebrasO([0,1]N) and O([0,1]n) satisfy preciselythe
same equations — i.e., in both cases the corresponding logic is intuitionistic logic
— regardless of the fact that one cube has strictly larger topological dimension
than the other. However, if we consider the smaller Heyting algebras of open
subpolyhedra ofthe two cubes, then bd is validin [0,1]n andis refuted in [0,1]N.
n
Restriction to a class of tame, geometric subsets of Euclidean space such as the
polyhedraofourpaperthusallowsus toexpressthe dimensionofEuclideanspaces
by means of intuitionistic logic and Heyting algebras.
2. Preliminaries
We assume familiarity with intuitionistic logic and Heyting algebras. A few
standard references are [4, 18, 19, 10]. In this section we recall what we need. On
the other hand, we assume rather less about PL topology. All needed definitions
and results are recalled in detail in this section. A few standard references are
[31, 15, 14, 20, 30].
‘Distributive lattice’ means ‘bounded distributive lattice’; homomorphisms are
topreserveboththemaximum(⊤)andtheminimum(⊥)element. Wewrite∧and
∨ for meets and joins, and write → and ¬ for Heyting implication and negation.
2.1. Posets, frames and p-morphisms. Wedenotethepartialorderrelationon
any poset by 6, unless otherwise specified. Given any poset A and any a ∈A, we
set
↑a:={x∈A|a6x},
↓a:={x∈A|x6a}.
An upper set in A is a subset U ⊆A closed under ↑: if a∈A satisfies a∈U, then
↑a⊆U. Similarly,a lower set in Ais asubset closedunder ↓. Achain is a totally
ordered set. A chain in A is a subset C ⊆ A that is a chain when equipped with
the order inherited from A. We define the depth of A to be
depA:=sup{|C|−1 | C ⊆A is a chain in A}∈N∪{∞}.
4 N.BEZHANISHVILI,V.MARRA,D.MCNEILL,ANDA.PEDRINI
If A and B are posets, a p-morphism from A to B is an order-preserving function
f: A→B that commutes with ↑: for each a∈A,
f[↑a]=↑f(a).
Here and throughout, f[·] denotes direct image under the function f. Similarly,
f−1[·] will denote inverse image under the function f.
An(intuitionistic Kripke) frame isjustaposet. Itisrooted ifithasaminimum.
Any frame A gives rise to a Heyting algebra. First, set
UpA:={U ⊆A|U is an upper set in A}.
Under the inclusion order, UpA is a complete distributive lattice; arbitrary meets
and joins are provided by set-theoretic unions and intersections. Hence the meet
operation has an adjoint, the uniquely determined implication of UpA that makes
it into a Heyting algebra.4 For later use in the paper, we also prepare the dual
notation
LoA:={L⊆A|L is lower set in A}.
AsforUpA,wewillalwaysregardLoAasacompletedistributivelatticeunderthe
inclusion order. LoA has a uniquely determined co-Heyting algebraic structure.5
Conversely,we can associate a poset to any Heyting algebra H. Set
SpecH :={F ⊆H |F is a prime filter}.
Here, we mean that F is a prime filter of the underlying distributive lattice of H.
Equipping SpecH with the inclusion order, we obtain a poset.
The Heyting algebras of the form UpA, as A ranges over all finite posets, are
precisely the finite Heyting algebras. To see this, given a Heyting algebra H, we
consider the Stone map:
·: H −→UpSpecH (1)
h∈H 7−→h:={p∈SpecH |h∈p}.
b
The following goes back to [8].
b
Lemma 2.1. For any finite Heyting algebra H, the Stone map (1) is an isomor-
phism of Heyting algebras.
Proof. For detailed proofs see [10, Sec. 8.4] and [25]. (cid:3)
Withtheaboveinplace,amodernstatementofapartofJa´skowski’sresultcited
in the Introduction is:
Lemma 2.2 (The finite modelproperty). The equational class of Heyting algebras
is generated by the finite Heyting algebras: any Heyting algebra is a homomorphic
imageofasubalgebraofaproductofHeytingalgebrasoftheformUpA,asAranges
over all finite posets.
Proof. See e.g. [10, Thms. 2.57 and 7.21]. (cid:3)
Remark 2.3. One can restrict the class of finite posets featuring in Lemma 2.2 in
various ways. Thus, Ja´skowskiexhibited a specific recursive sequence of posets. It
is also known,for instance, that the classof allfinite trees (=rootedframesT such
that ↓t is a chain for each t∈T) suffices, see e.g., [10, Cor. 2.33 and Ex. 2.17]. In
this paper we only need the generalform of the result as stated in Lemma 2.2. (cid:3)
4SeeSubsection2.4forthegeneralisationofthisconstructiontoalltopological spaces.
5AswellasaHeytingonethatwillnotbeusedinthispaper.
TARSKI’S THEOREM AND POLYHEDRA 5
2.2. Finite Esakia duality. Lemma 2.1 can be lifted to a contravariant equiva-
lence of categories between Heyting algebras and Esakia spaces [11]. In the finite
case of interest here, topology can and will be dispensed with. Given a homomor-
phism of finite Heyting algebras h: H →K, set
Spech: SpecK −→SpecH
p∈SpecK 7−→h−1[p]∈SpecH.
Dually, given a p-morphism of posets f: A→B, set
Upf: UpB −→UpA
U ∈UpB 7−→f−1[U]∈UpA.
Let now HAf and Posf denote the categories of finite Heyting algebras and their
homomorphisms, and of finite posets and p-morphisms, respectively. Then the
above defines functors
Spec: HAf −→Posofp,
Up: Posf −→HAofp.
(We areindicating by Cop the categoryopposite to the categoryC, asis standard.)
Lemma 2.4 (Esakia duality, finite case). The functors Spec and Up are an equiv-
alence of categories.
Proof. See [10, Exs. 7.5, 7.6 and Sec. 8.5] and [25]. (cid:3)
Remark 2.5. As with all duality results, Lemma 2.4 provides a dictionary between
notions in HAf and notions in Posf. For example, one shows that a surjective p-
morphism of finite posets dualises to an injective homomorphism of finite Heyting
algebras, i.e. to a Heyting subalgebra, and conversely. We do not dwell on the
details of such translations, and use them whenever needed in the sequel. (cid:3)
2.3. Bounded depth. Through the equations corresponding to the formulæ bd
d
oftheIntroduction,onecanexpressequationallytheanalogueforHeytingalgebras
of the Krull dimension of commutative rings.6
Lemma 2.6. For any non-trivial Heyting algebra H and each d∈N, the following
are equivalent.
(i) The longest chain of prime filters in H has cardinality d+1.
(ii) depSpecA=d.
(iii) H satisfies the equation bdd = ⊤, and fails each equation bdd′ = ⊤ with
16d′ <d.
Proof. See [10, Prop. 2.38 and Table 9.7] and [7]. (cid:3)
2.4. Heyting and co-Heyting algebras of open and closed sets. The open
(closed) sets of a topological space provide important examples of (co-)Heyting
algebras. For background on co-Heyting algebras we refer to [23, §1 and passim],
wherethesestructureswerefirstaxiomatisedequationally,andsystematicallyinves-
tigatedunderthe nameof‘Brouwerianalgebras’. We write¬to denote co-Heyting
negation, and ← to denote co-Heyting implication.7
6For recent related literature see [7], where a Krull dimension is defined for any topologi-
cal space and is used in obtaining fine-grained topological completeness results for modal and
intermediatelogics.
7McKinsey’sandTaski’soriginalnotations were¬and−.,respectively.
6 N.BEZHANISHVILI,V.MARRA,D.MCNEILL,ANDA.PEDRINI
IfX isanytopologicalspace,wewriteO(X)foritscollectionofopenssets. Then
O(X) is a complete distributive lattice, bounded above by ⊤ := X and below by
⊥:=∅, with joins given by set-theoretic unions and meets given by
F :=int F
foranyfamilyF ofopensubsetso^fX,wherei\ntdenotestheinterioroperatorofthe
giventopologyonX. ThereforeO(X)hasexactlyonestructureofHeytingalgebra
compatiblewithitsdistributive-latticestructure;namely,foranyU,V ∈O(X)the
Heyting implication is given by
U →V := {O∈O(X)|U ∩O⊆V}=int((X \U)∪V)). (2)
In particular, the Hey[ting negation is given by
¬U :=U →⊥=int(X \U).
Dually, the family C (X) of closed sets of X is a complete distributive lattice,
bounded above by ⊤:=X and below by ⊥:=∅, with meets given by set-theoretic
intersections and joins given by
F :=cl F
for any family F of closed subse_ts of X, w[here cl denotes the closure operator
of the given topology on X. Therefore C (X) has exactly one structure of co-
Heyting algebra compatible with its distributive-lattice structure; namely, for any
C,D ∈C (X) the co-Heyting implication is given by
C ←D := {K ∈C (X)|C ⊆D∪K}=cl(C\D). (3)
In particular, the co-Heyt\ing negation is given by
¬D :=⊤←D =cl(X \D).
Remark 2.7. AllourresultsinthispaperhaveversionsforHeytingandco-Heyting
algebras. We stressed the Heyting versionin the Introduction, as this relates most
directly to inutitionistic logic. However, we will see below that it is at times con-
venient in proofs to establish the co-Heyting version of the results first, because it
is traditional in simplicial topology to work with closed simplices and polyhedra.
ProofsforthecorrespondingHeytingversionsareobtainedthroughdualarguments,
which we sometime omit. (cid:3)
2.5. Polyhedra: basic notions. An affine combination of x ,...,x ∈ Rn is an
0 d
element d r x ∈Rn, where r ∈R and d r =1. If, additionally, r >0 for
i=0 i i i i=0 i i
each i∈ {0,...,d}, d r x is a convex combination. Given any subset S ⊆Rn,
P i=0 i i P
the convex hull of S, written convS, is the collectionof allconvex combinations of
P
finitesubsetsofS. ThenS isconvex ifS =convS,andapolytope ifS =convV for
afinitesetV ⊆Rn. Apolyhedron inRn isanysubsetthatcanbewrittenasafinite
union of polytopes. The union over an empty index set is allowed, so that ∅ is a
polyhedron. Anypolyhedronisclosedandbounded,hencecompact. IfP ⊆Rn isa
polyhedron, by anopen polyhedron in P we mean the complement of a polyhedron
which is included in P. The points x ,...,x ∈Rn are affinely independent if the
0 d
vectors x −x ,x −x ,...,x −x are linearly independent, a condition which is
1 0 2 0 d 0
invariantunderpermutationsoftheindexset{0,...,d}. Asimplex inRn isanon-
empty8 subset of the form σ :=convV, where V :={x ,...,x } is a set of affinely
0 d
independent points. Then V is the uniquely determined such affinely independent
set [20, Proposition 2.3.3], and σ is a d-simplex with vertices x ,...,x . A face
0 d
of the simplex σ is the convex hull of a non-empty subset of V, and thus is itself
8Itisexpedient inthispapernottoregard∅asasimplex.
TARSKI’S THEOREM AND POLYHEDRA 7
a d′-simplex for a uniquely determined d′ ∈ {0,...,d}. Hence the 0-faces of σ are
precisely its vertices.
We write
σ =x ···x , σ 4τ, and σ ≺τ
0 d
to indicate that σ is the d-simplex whose vertices are x ,...,x , that σ is a face of
0 d
τ, and that σ is a proper (i.e. 6= τ) face of τ, respectively. If σ = x ···x ∈ Rn,
0 d
the relative interior of σ, denoted relintσ, is the topological interior of σ in the
affinesubspaceofRn spannedby9σ. (Thus,therelativeinteriorofa0-dimensional
simplex — a point — is the point itself.) To rephrase through coordinates, note
that by the affine independence of the vertices of σ, for each x ∈ σ there exists
a unique choice of r ∈ R with x = d r x and r > 0, d r = 1. The
i i=0 i i i i=1 i
r ’saretraditionallycalledthebarycentric coordinates ofx. Thenrelintσ coincides
i
P P
withthesubsetofσ ofthosepointsx∈σwhosebarycentriccoordinatesarestrictly
positive. Notethatclrelintσ =σ,theclosurebeingtakenintheambientEuclidean
space Rn. In the rest of this paper, for any set S ⊆Rn we use the notation
clS (4)
to denote the closure of S in the ambient Euclidean space Rn. Observe that if
P ⊆Rn is a polyhedron and S ⊆P, then the closure of S in the subspace P of Rn
agrees with clS, because P is closed in Rn.
2.6. Polyhedra: the Triangulation Lemma.
Definition 2.8 (Triangulation). A triangulation10 is a finite set Σ of simplices in
Rn satisfying the following conditions.
(1) If σ ∈Σ and τ is a face of σ, then τ ∈Σ.
(2) If σ,τ ∈Σ, then σ∩τ is either empty, or a common face of σ and τ.
The support, or underlying polyhedron, of the triangulation Σ is
|Σ|:= Σ ⊆Rn.
One also says that Σ triangulates the[subset |Σ| of Rn. A subtriangulation of
the triangulation Σ is any subset ∆ ⊆ Σ that is itself a triangulation. This is
equivalentto the conditionthat ∆ be closedunder taking faces — i.e.satisfies just
(1) in Definition 2.8 — for then (2) follows [20, Proposition 2.3.6]. By the vertices
of Σ we mean the vertices of the simplices in Σ.
Observe that a subtriangulation of Σ is precisely the same thing as a lower
set of Σ, the latter being regarded as a poset under inclusion. This fact will be
heavily exploited below, cf. in particular Section 4. The following standard fact
makesprecisetheideathatatriangulation|Σ|providesafinitarydescriptionofthe
triangulated space Σ.
Lemma 2.9. If Σ is a triangulation, for each x∈|Σ| there is exactly one simplex
σx ∈Σ such that x∈relintσ.
Proof. See [20, Proposition 2.3.6]. (cid:3)
9Recall that the affine subspace spanned by a subset S ⊆ Rn is the collection of all affine
combinations of finite subsets of S, or equivalently, the intersection of all affine subspaces of Rn
containingS.
10Also known as (geometric) simplicial complex. Note that the empty triangulation ∅ is
allowed.
8 N.BEZHANISHVILI,V.MARRA,D.MCNEILL,ANDA.PEDRINI
In light of Lemma 2.9, in the sequel we adopt the notation σx without further
comment; the simplex σx is called the carrier of x (in Σ).
AnysubsetofRn thatadmitsatriangulation,beingafiniteunionofsimplices,is
evidentlyapolyhedron. Theratherlesstrivialconverseistrue,too,inthefollowing
strong sense.
Lemma2.10(TriangulationLemma). GivenfinitelymanypolyhedraP,P ,...,P
1 m
in Rn with P ⊆P for each i∈{1,...,m}, there exists a triangulation Σ of P such
i
that, for each i∈{1,...,m}, the collection
Σ :={σ ∈Σ|σ ⊆P }
i i
is a triangulation of P , i.e. |Σ |=P .
i i i
Proof. [30, Theorem 2.11 and Addendum 2.12]. (cid:3)
The Triangulation Lemma is the fundamental tool in this paper. Recall from
the Introduction that Sub P and Sub P denote the collections of polyhedra and
c o
open polyhedra in P, respectively. Here is a first consequence11 of Lemma 2.10.
Corollary 2.11. For any polyhedron P ⊆ Rn, both Sub P and Sub P are dis-
c o
tributive lattices (under set-theoretic intersections and unions) bounded above by P
and below by ∅. (cid:3)
Proof. Given polyhedra A,B ⊆P, by Lemma 2.10 there is a triangulation Σ of P
along with two subtriangulations Σ ,Σ with A = |Σ | and B = |Σ |. Then the
A B A B
triangulation ∆ := Σ ∩Σ triangulates A∩B. Indeed, obviously |∆| ⊆ A∩B.
A B
Conversely, if x ∈ A∩B then there are σ ∈ Σ ,σ ∈ Σ with x ∈ σ and
A A B B A
x ∈ σ . Setting τ := σ ∩σ , we have x ∈ τ 6= ∅ and τ ∈ Σ. Since τ is a face of
B A B
σ ∈Σ , τ ∈Σ . Similarly, τ ∈Σ . Hence τ ∈∆, and A∩B ⊆|∆|. Similarly, it
A A A B
iselementarythatthetriangulation∇:=Σ ∪Σ triangulatesA∪B. Itisobvious
A B
that P and ∅ are the upper and lower bounds of Sub P. The statements about
c
Sub P follow at once by taking complements. (cid:3)
o
In Subsection 3.1 we shall strengthen Corollary2.11 to the effect that Sub P is
o
a Heyting subalgebra of the Heyting algebra O(P).
2.7. Polyhedra: dimension theory. The (affine) dimension ofa d-simplex σ =
x ···x in Rn is the linear-space dimension of the affine subspace of Rn spanned
0 d
by σ, and that dimension is precisely d because of the affine independence of the
vertices of σ. The (affine) dimension of a nonempty polyhedron P in Rn is the
maximumofthe dimensions ofallsimplicescontainedinP;ifP =∅,its dimension
is −1. We write dimP for the dimension of P. Given a triangulation Σ in Rn, the
(combinatorial) dimension of Σ is
dimΣ:=max{d∈N| there exists σ ∈Σ such that σ is a d-simplex.}
Again, the dimension of an empty triangulation is −1. Everything in the lemma
that follows is of course classical.
Lemma 2.12. For any polyhedron ∅6=P ⊆Rn and every d∈N, the following are
equivalent.
(i) dimP =d.
(ii) There exists a triangulation Σ of P such that dimΣ=d.
(iii) All triangulations Σ of P satisfy dimΣ=d.
(iv) The Lebesgue covering dimension [27, Definition 3.1.1] of the topological
space P is d.
11Cf.[20,Proposition2.3.6(d)].
TARSKI’S THEOREM AND POLYHEDRA 9
Proof. With the Triangulation Lemma 2.10 available, the equivalences (i) ⇔ (ii)
⇔ (iii) follow from linear algebra. The equivalence (i) ⇔ (iv) is, in essence, the
Lebesgue Covering Theorem [16, Theorem IV 2]. (cid:3)
3. The locally finite Heyting algebra of a polyhedron
Throughoutthissectionwefixn∈NalongwithapolyhedronP ⊆Rn. Weshall
study the distributive lattice Sub P (Corollary 2.11). We begin by proving that
o
Sub P is in fact a Heyting algebra. We then prove that Sub P is always locally
o o
finite.
3.1. The Heyting algebra of open subpolyhedra. Letusrecordawell-known,
elementary observation on relative interiors for which we know no convenient ref-
erence.
Lemma 3.1. Let Σ be a triangulation in Rn, let τ = x ···x be a simplex of Σ,
0 d
and let x ∈ relintτ. Then no proper face σ ≺ τ contains x. Hence, in particular,
the carrier σx of x in Σ is the inclusion-smallest simplex of Σ containing x.
Proof. There are r ,...,r ∈ (0,1] such that x = d r x and d r = 1. Let
0 d i=0 i i i=0 i
ρ := x ···x x ···x . Clearly ρ ≺ τ for each i ∈ {0,...,d}, and for each
i 0 i−1 i+1 d i
P P
σ ≺ τ there exists i ∈ {0,...,d} such that σ 4 ρ . Hence, if we assume by way of
i
contradictionthatx∈σ ≺τ,thenx∈ρ forsomei∈{0,...,d};sayx∈ρ . Then
i 0
x = d s x , for some s ,...,s ∈ [0,1] such that d s = 1. It follows that
i=1 i i 1 d i=1 i
r = d (s −r ), and so
0 Pi=1 i i P
d d d d
P
0=x−x= s x − r x = (s −r )x −r x = (s −r )(x −x ).
i i i i i i i 0 0 i i i 0
i=1 i=0 i=1 i=1
X X X X
Since r > 0, there must be i ∈ {1,...,d} such that s −r 6= 0, contradicting the
0 i i
affine independence of x ,...,x . (cid:3)
0 d
The next lemma is the key fact of this subsection.12
Lemma 3.2. Let P and Q be polyhedra in Rn with Q ⊆ P, and suppose Σ is a
triangulation of P such that
Σ :={σ ∈Σ|σ ⊆Q}
Q
triangulates Q. Define
• C :=cl(P \Q),
• Σ :={σ ∈Σ|σ ⊆C}, and
C
• Σ∗ :={σ ∈Σ|There exists τ ∈Σ\Σ such that σ 4τ}.
Q
Then
(1) Σ =Σ∗, and
C
(2) |Σ |=|Σ∗|=C.
C
In particular, C is a polyhedron.
Proof. We first show that Σ∗ triangulates C, that is:
|Σ∗|:= Σ∗ =C. (*)
Toshow|Σ∗|⊆C,letσ ∈Σ∗,andpick[τ ∈Σ\Σ suchthatσ 4τ. We provethat
Q
relintτ ⊆ P \Q. For, if x ∈ relintτ, by Lemma 3.1 there are no simplices σ ∈ Σ
such that x ∈ σ ≺ τ. Then, by definition of triangulation, for any simplex ρ ∈ Σ,
x∈ρ entailsτ 4ρ. Hence no simplex ofΣ containsx, orequivalently, x6∈Q and
Q
therefore relintτ ⊆P \Q.
12Cf.[20,Proposition2.3.7].
10 N.BEZHANISHVILI,V.MARRA,D.MCNEILL,ANDA.PEDRINI
Now, it is clear that any simplex τ satisfies τ = clrelintτ. It follows that
σ ⊆τ =clrelintτ ⊆cl(P \Q), and thus |Σ∗|⊆C as was to be shown.
Conversely, to show C ⊆|Σ∗|, let x∈C. Since C is the closure of P \Q in Rn,
there exists a sequence {xi}i∈N ⊆ P \Q that converges to x. Clearly the carrier
σxi of x in Σ lies in Σ\Σ , for all i∈N. Since Σ\Σ is finite, there must exist
i Q Q
a simplex τ ∈ Σ\ΣQ containing infinitely many elements of {xi}i∈N. Then there
exists a subsequence of {xi}i∈Nthat is contained in τ and converges to x. Since τ
is closed, x∈τ, and therefore x∈|Σ∗| as was to be shown.
This establishes (*). It now suffices to prove (1). For the non-trivial inclusion
Σ ⊆Σ∗, let σ ∈Σ be such that σ ⊆C, and pick β ∈relintσ. There is a sequence
C
{xi}i∈N ⊆ P \Q converging to β ∈ σ. Since each xi is in some simplex of Σ\ΣQ
and Σ is finite, there must exist a simplex τ ∈Σ\Σ containing a subsequence of
Q
{xi}i∈N that converges to β. Since τ is closed, β ∈τ. But by Lemma 3.1, σβ =σ,
so that σ ⊆τ and σ ∈Σ∗. (cid:3)
Corollary 3.3. Given polyhedra Q ,Q in Rn, theset cl(Q \Q )is apolyhedron.
1 2 2 1
Proof. Observe that Q \Q = Q \(Q ∩Q ) and apply Corollary 2.11 together
2 1 2 1 2
with Lemma 3.2to the set P :=conv(Q ∪Q ), which clearlyis a polyhedron. (cid:3)
1 2
Corollary 3.4. The lattice Sub P of is closed under the co-Heyting implication
c
(3) of C (P). Dually, the lattice Sub P is closed under the Heyting implication (2)
o
of O(P).
Proof. The first statement is an immediate consequence of Corollary 3.3. The
second statement follows by dualising. (cid:3)
3.2. Local finiteness through triangulations. Having established that Sub P
o
is a Heyting subalgebra of O(X), we infer an important structural property of
Sub P,localfiniteness. Forthis,wefirstidentifytheclassofsubalgebrasofSub P
o o
that corresponds to triangulations of P. These algebras will have a central rˆole in
the sequel, too.
Definition 3.5 (Σ-definable polyhedra). For any triangulation Σ in Rn, we write
P (Σ) for the sublattice ofC (|Σ|) generatedby Σ, andP (Σ) for the sublattice of
c o
O(|Σ|) generated by {|Σ|\C | C ∈ P (Σ)}. We call P (Σ) the set of Σ-definable
c c
polyedra, and P (Σ) the set of Σ-definable open polyedra.
o
Note that we have
P (Σ)={C ⊆Rn |C is the union of some subset of Σ}.
c
Lemma 3.6. For any triangulation Σ of P, P (Σ) is a co-Heyting subalgebra of
c
Sub P. Dually, P (Σ) is a Heyting subalgebra of Sub P.
c o o
Proof. For any ∅=6 C,D ∈P (Σ), it follows immediately by the assumptions that
c
C and D are triangulated by the collection of simplices of Σ contained in C and
D, respectively. Hence C ← D := cl(C\D) = |Σ∗| = Σ∗ by Corollary 3.3 and
Lemma 3.2, where Σ∗ is the appropriate subset of Σ as per Lemma 3.2. Thus
S
C ←D ∈P (Σ). (cid:3)
c
Corollary 3.7. Let H be the co-Heyting subalgebra of Sub P generated by finitely
c
many polyhedra P ,...,P ⊆ P. Let further Σ be any triangulation of P that
1 m
triangulates each P , i∈{1,...,m}. Then H is a co-Heyting subalgebra of P (Σ).
i c
In particular, H is finite. Dually for the Heyting subalgebra of Sub P generated by
o
P \P , i∈{1,...,m}.
i